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Math Help - Poisson Distribution question

  1. #1
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    Poisson Distribution question

    Hi
    Can someone tell me if my answers are correct?
    A machine makes a special type of lining brick at the rate of 25 per hour. Overall about 5% of the bricks are
    defective. Calculate:
    The probability that no defective bricks are produced in any one hour?
    Now this is a poisson distributions so i used the calculated by using the calculator function
    poissonpdf(1.25,0) = 0.2865
    however answer is 0.2774

    the probability that two defective bricks are produced in any one hour?
    poissonpdf(1.25,2) = 0.2238
    however answer is 0.2305

    the probability that at least one defective brick is produced in the next hour?
    not sure about this one

    P.S
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Poisson Distribution question

    Quote Originally Posted by Paymemoney View Post
    Hi
    Can someone tell me if my answers are correct?
    A machine makes a special type of lining brick at the rate of 25 per hour. Overall about 5% of the bricks are
    defective. Calculate:
    The probability that no defective bricks are produced in any one hour?
    Now this is a poisson distributions so i used the calculated by using the calculator function
    poissonpdf(1.25,0) = 0.2865
    however answer is 0.2774

    the probability that two defective bricks are produced in any one hour?
    poissonpdf(1.25,2) = 0.2238
    however answer is 0.2305

    the probability that at least one defective brick is produced in the next hour?
    not sure about this one

    P.S
    The probability of k events in n trials has binomial distribution, even if for 'rare events' the Poisson distribution [computationallly less problematic...] gives accetable precision. The probability to have k defective briks in a stock of n, if p is the probability of a single defective brick, is...

    P(k,n)= \binom{n}{k}\ p^{k}\ (1-p)^{n-k} (1)

    If p=.05 and n=25 the (1) supplies P(0,25)= .277389573... and P(2,25)=.23051765079.... The probability of least one defective brick in one hour is of course 1-P(0,25)= .722610426878...

    Kind regards

    \chi \sigma
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  3. #3
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    Re: Poisson Distribution question

    but doesn't any events that occur at random with a constant average per some unit such as time and length or area is modeled as a Poisson distribution?
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  4. #4
    MHF Contributor chisigma's Avatar
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    Re: Poisson Distribution question

    Quote Originally Posted by Paymemoney View Post
    but doesn't any events that occur at random with a constant average per some unit such as time and length or area is modeled as a Poisson distribution?
    The real probability distribution of such type of process is binomial, even if in many pratical situations the Poisson distribution gives an excellent approximation. Binomial and Poisson distributions give pratically the same results when is \lambda = p\ n <<1. In Your case is \lambda= p\ n = 1.25, so that the results are slighly different. Here You can observe the values of...

    P_{b} (k,n) = \binom {n}{k}\ p^{k}\ (1-p)^{n-k} (1)

    P_{p}(k,\lambda)= \frac{\lambda^{k}\ e^{-\lambda}}{k!} (2)

    ... computed for k from 0 to 9 with \lambda= p\ n= 1.25...

    k=0\ ,\ P_{b}= .2773895\ ,\ P_{p}= .2865048

    k=1\ ,\ P_{b}= .3649863\ ,\ P_{p}= .358131

    k=2\ ,\ P_{b}= .2305176\ ,\ P_{p}= .2238318

    k=3\ ,\ P_{b}= .09301589\ ,\ P_{p}= .09326328

    k=4\ ,\ P_{b}= .02692565\ ,\ P_{p}= .02914477

    k=5\ ,\ P_{b}= .0059519866\ ,\ P_{p}= .0072861937

    k=6\ ,\ P_{b}= .001044208\ ,\ P_{p}= .0015177957

    k=7\ ,\ P_{b}= .0001491726\ ,\ P_{p}= .00027106375

    k=8\ ,\ P_{b}= 1.76651758\ 10^{-5}\ ,\ P_{p}= 4.23537119\ 10^{-5}

    k=9\ ,\ P_{b}= 1.75618707\ 10^{-7}\ ,\ P_{p}= 5.88246\ 10^{-6}

    The most evident difference from the two is the fact that, for 'large' value of k, the P_{b} decreases much steeper than the P_{p} and the reason of that is obvious...

    Kind regards

    \chi \sigma
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